RobustMultitaskDiffusionNormalizedM-estimate SubbandAdaptiveFilteringAlgorithmOverAdaptive Networks

2025-05-03 0 0 2.49MB 19 页 10玖币

Robust Multitask Diffusion Normalized M-estimate

Subband Adaptive Filtering Algorithm Over Adaptive

Networks

Wenjing Xu1, Haiquan Zhao1.*, Shaohui Lv1

Abstract: In recent years, the multitask diffusion least mean square (MD-LMS) algorithm has been extensively applied in the

distributed parameter estimation and target tracking of multitask network. However, its performance is mainly limited by two

aspects, i.e, the correlated input signal and impulsive noise interference. To overcome these two limitations simultaneously, this

paper firstly introduces the subband adaptive filter (SAF) into the multitask network. Then, a robust multitask diffusion

normalized M-estimate subband adaptive filtering (MD-NMSAF) algorithm is proposed by solving the modified Huber function

based global network optimization problem in a distributed manner, which endows the multitask network strong decorrelation

ability for correlated inputs and robustness to impulsive noise interference, and accelerates the convergence of the algorithm

significantly. Compared with the robust multitask diffusion affine projection M-estimate (MD-APM) algorithm, the computational

complexity of the proposed MD-NMSAF is greatly reduced. In addition, the stability condition, the analytical expressions of the

theoretical transient and steady-state network mean square deviation (MSD) of the MD-NMSAF are also provided and verified

through computer simulations. Simulation results under different input signals and impulsive noise environment fully demonstrate

the performance advantages of the MD-NMSAF algorithm over some other competitors in terms of steady-state accuracy and

tracking speed.

Keywords: Distributed Estimation, Multitask Network, Subband Adaptive Filter, Impulsive Noise, M-estimate Function

1. Introduction

Distributed adaptive estimation is an important technique for signal processing within a network, which performs specific tasks

through continuous learning and adaptation of interconnected nodes in network. It has attracted extensive research due to its

reliability, scalability, and resource efficiency [1]-[3]. Previous research on distributed estimation mainly focused on the

distributed single-task network, the whole network only needs to estimate a single parameter vector. In this work, the distributed

estimation of the cluster multitask network is considered, which is different from the single-task network in that there are multiple

unknown parameter vectors need to be estimated. All nodes in the multitask network are divided into different clusters, each

cluster has its own task (parameter vector to be estimated) [4]-[7].

1Wenjing Xu, Haiquan Zhao and Shaohui Lv are with the Key Laboratory of Magnetic Suspension Technology and Maglev Vehicle, Ministry of Education, and the

School of Electrical Engineering, Southwest Jiaotong University, Chengdu,610031,China.

*Corresponding author

E-mail:wenjingxv@126.com; hqzhao@home.swjtu.edu.cn; shaohuilv_swjtu@126.com;

In recent years, the multitask diffusion LMS (MD-LMS) algorithm has been well applied to the parameter estimation and target

tracking of the multitask networks [6]. However, its performance is mainly limited by two aspects: On the one hand, its

convergence speed largely depends on the characteristics of the input signal. Specifically, the large eigenvalue spread of the

correlation matrix of the correlated input signal will lead to slow convergence of the algorithm. On the other hand, the optimality

of the algorithm based on the MMSE criterion depends on the Gaussian noise assumption to a large extent. When the

measurement noise does not conform to the Gaussian distribution, the performance of the MD-LMS algorithm will deteriorate

significantly.

In view of the former defect, reference [8] proposes the multitask diffusion affine projection algorithm (MD-APA) to improve

the convergence speed of the algorithm under the correlated inputs. In fact, the MD-APA is a generalization of the normalized

version of MD-LMS in the time domain. For the latter defect, on the basis of MD-APA, the multitask diffusion affine projection

M-estimate (MD-APM) algorithm is proposed by applying the M-estimate function to remove outliers [9], [10]-[13]. The

multitask diffusion affine projection maximum correntropy criterion (MD-APMCC) algorithm is also proposed by introducing the

maximum correntropy criterion (MCC) into the MD-APA [14], [15], which is insensitive to large outliers. These algorithms are all

robust to impulsive noise interference.

Unfortunately, the above robust multitask algorithms are all based on the AP strategy, which not only improve the convergence

speed of the algorithm, but also greatly increase the computational complexity. Inspired by the subband adaptive filter (SAF)

[16]-[22], we extend the normalized version of the MD-LMS algorithm to the subband domain, and introduce the M-estimate

function with the property of removing outliers, and propose a robust multitask diffusion normalized M-estimate subband adaptive

filtering (MD-NMSAF) algorithm. The proposed MD-NMSAF algorithm inherits the strong decorrelation ability of SAF,

improves the convergence speed of the algorithm under correlated inputs and endows the distributed network robustness to

impulsive noise. Furthermore, compared with the MD-APA, the structure of SAF can greatly reduces the computational

complexity of the MD-NMSAF algorithm. The main contributions of this paper are summarized as follows:

1) The cluster multitask diffusion algorithm is extended to the subband domain, and the M-estimate function is introduced to

propose a robust multitask diffusion SAF algorithm, i.e, MD-NMSAF algorithm, which has the strong decorrelation ability

for correlated inputs and robustness to impulsive noise interference.

2) The mean and mean square stability of the MD-NMSAF algorithm are analyzed in impulsive noise environment, and the

stability conditions are also given. Then, the analytical expressions of the theoretical transient and steady-state network

network mean square deviation (MSD) are provided.

3) The computational complexity of the proposed MD-NMSAF algorithm with some other multitask diffusion algorithms are

summarized in the form of table, and the computational advantages of the MD-NMSAF algorithm over the AP based

multitask diffusion algorithm are explained in detail.

4) The influences of the step size and subband number on the performance of MD-NMSAF are studied, and the validity and

accuracy of the mean square stable step size upper bound, theoretical transient and steady-state analysis results are verified

under different input signals.

5) The convergence and tracking performance between the proposed MD-NMSAF algorithm and some other multitask

diffusion algorithms are compared in different input signals, the superiority of the MD-NMSAF algorithm is verified.

The remainder of this paper is organized as follows. In section 2, the MD-NMSAF algorithm is derived in detail. In section 3,

the mean and mean-square stability, theoretical transient and steady-state performance and computational complexity of the

proposed MD-NMSAF algorithm are analyzed. Computer simulations are provided in Section 4. Finally, the conclusion is given

in Section 5.

2. The proposed algorithm

Consider a clustered multitask network consists of Nnodes, the nodes on this network are divided into Lclusters (Ltasks).

Node khas access to the reference signal

( )

d t

and input signal vector

( )

ktu

, both of them are related by the following linear

model:

( ) ( ) ( ) u w

k k k k

d t t v t

(1)

where the

1M

vector

is the target parameter vector of node k, and

( )

v n

denotes the measurement noise at node k. The

target parameter vectors of nodes in the same cluster

, 1, 2,

ll LC

are the same, i.e.,

* * ,l

w wC

for

l

. In addition, it is

assumed that there are some similarities between the target parameter vectors of neighboring clusters [4].

Fig.1. Multiband structure of subband adaptive filter.

In order to accelerate the convergence of the MD-LMS algorithm under highly correlated input, we introduce the SAF into the

multitask network and reduces the correlation of input signal by subband dividing and extracting [16]. Fig.1 shows the

multiband-structured SAF with

subbands. For each node k, the reference signal

( )

d t

and input signal vector

( )

ktu

are

partitioned into

band-dependent signals

,( )

k i

d t

and

,( )

k i tu

by the analyzing filter bank

{ ( ), 0,1, , 1}

i D

H z i N 

respectively. The subband output signal

,( )

k i

y t

is obtained by inputting the subband signal

,( )

k i tu

into the SAF with tap-weight

vector

,1 ,2 ,

( ) [ ( ), ( ), , ( )]T

k k k k M

n w n w n w nw

. Then,

,( )

k i

d t

and

,( )

k i

y t

are critically decimated to

, ,

( ) ( )

k iD k i D

d n d nN

and

, ,

( ) ( )

k iD k i D

y n y nN

, respectively. The variables tand nrepresent the index of the original and decimated sequences, respectively.

Therefore, the decimated subband error signal

,( )

k iD

e n

can be expressed as

, , ,

, ,

( ) ( ) ( )

k iD k iD k iD

k iD k i k

e n d n y n

d n n n

 

  u w

(2)

where

, , , ,

( ) ( ), ( 1),..., ( 1) T

k i k i D k i D k i D

n nN nN nN M

 

   

 

u u u u

. Similar to the MD-LMS algorithm [6], to solve a multitask

network global optimization problem in a distributed manner, the cost function must rely on the local interaction of measurement

data within neighborhoods and the similarity relationship between adjacent tasks. At the same time, after introducing the

M-estimate function, we can construct the following local cost function

loc loc

( )

, , 20

222

( ) 0 \ ( ) ( )

( ) ( ) ( )

( ) ( )

2 2

( )







   



 



 

    



   







k k k

k k k k l l

l k

Nr iD r i k

k,r k,l k l m k k m

r k i l k m k

r i

J J J

d n n

c b



w w w

u w w w w w

N C

N C N C N C

(3)

where

represents the set of neighbor nodes of node k(including itself),



is the set that

excluding node k, the

symbol



denotes the set intersection,

is the set difference, and

( )kC

denotes the cluster where node kis located.

k,r

and

k,l



are non-negative intra-task and inter-task combination coefficients, respectively.

,m k

is a non-negative coefficient, please

refer to [2] for details.



is called the regularization strength [6].

is the target parameter vector of node m. Note that the

M-estimate function

[ ]





adopted in this paper is the modified Huber function [10], given by

, , ,

( ) 2, ( ) ( )

[ ( )]

( ) 2, other











k iD k iD k iD

k iD

e n e n n

e n





(4)

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RobustMultitaskDiffusionNormalizedM-estimateSubbandAdaptiveFilteringAlgorithmOverAdaptiveNetworksWenjingXu1,HaiquanZhao1.*,ShaohuiLv1Abstract:Inrecentyears,themultitaskdiffusionleastmeansquare(MD-LMS)algorithmhasbeenextensivelyappliedinthedistributedparameterestimationandtargettrackingofmultitasknet...

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